Matrix Methods for Inhomogeneous Systems: Theory, Fundamental Matrix, Variation of Parameters
In this lesson, we learned about inhomogeneous systems of ODEs using matrix methods. The fundamental matrix, which is a matrix whose columns are two independent solutions, was introduced and its properties were discussed. One important property is that its determinant is never zero for any value of t, and the other property is that it satisfies a matrix differential equation, which is a way of saying that its two columns are solutions to the original system. Finally, we looked at an example of how to use these methods to solve inhomogeneous systems.
Matrix Methods for Inhomogeneous Systems: Theory, Fundamental Matrix, Variation of Parameters -- Lecture 28. Learn about inhomogeneous systems of ODEs by using matrices.
Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 29, 2008). License: Creative Commons BY-NC-SA.
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- Lecture Notes - Lecture Notes
- Linear Systems Notes - Linear Systems Notes
- Supplementary Notes - Chapter 24 Supplementary Notes written by Prof. Miller
- Problem Set 7 - Problem Set 7
- Problem Set 7 Solutions - Problem Set 7 Solutions
- What are Matrix Methods for Inhomogeneous Systems?
- What are Inhomogeneous Systems?
- What is the Fundamental Matrix?
- What is Variation of Parameters?
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Staff Review
An interesting discussion of inhomogeneous systems of linear differential equations by using matrices. The lecture is almost entirely theory, but overall, a good explanation of inhomogeneous systems. Wronskians come up again in this lecture as well, and the fundamental matrix is defined with its properties.
